# Paul Painlevé's books

We give below some extracts from reviews of some of Paul Painlevé's books. We give English versions of those reviews originally written in different languages.

**1. Les axiomes de la mécanique (1922), by Paul Painlevé.**

**1.1. Review by: Editors.**

*Revue de Métaphysique et de Morale*

**30**(2) (1923), 1-2.

This booklet from the "Masters of Scientific Thought" collection contains: a lesson on the axioms of mechanics, published for the first time in 1909; the reproduction of a communication made in 1904 to the Philosophical Society on the principle of causality; a "Note on the propagation of light", all preceded by an introduction, in which the author, comparing classical mechanics to that which follows from the theory of (restricted) relativity, warns against the errors of interpretation frequently encountered by Einstein's commentators.

It is the principle of causality and the notion of absolute motion which led to the discovery of the axioms of Mechanics. It is customary to say that modern mechanics was born on the day when the experimental method dethroned the 'a priori' methods of the scholastics. It is an inexact platitude, rightly points out M Painlevé. A priori ideas guided the creators of Mechanics, at least as much as precise experience, and these ideas were very close to scholastic ideas; they differed only in the modification, essential, it is true, made by Copernicus and his school to the principle of inertia. It is therefore at the school of Copernicus that we must trace the genesis of modern mechanics; Galileo, Kepler and Newton developed the Copernican ideas which enabled them to deduce from the rudimentary experiences and observations of Tycho Brahé all of Dynamics. The basic idea of the Copernicians was to rigorously subject the phenomena of movement to the principle of causality. The scholastics had the same claim. Like the Copernicans, they started from the notions of time, length and absolute movement (that is to say suitably identified), and like them they admitted that absolute movements satisfy the principle of causality. For both of them, the initial state of a material system determines its "future", that is to say its absolute movement. But, for scholastics, the initial state of the system is the positions of its elements, and nothing else; for the Copernicians, these are the positions and the velocities, at the initial instant. This divergence on the principle of inertia is enough to dig an abyss between the two doctrines.

Of course, the terms "copernicians" and "scholastics" do not mean here effectively constituted and officially opposing schools; they are simply a convenient way of distinguishing and opposing to each other the conception from which rational mechanics and the confused dynamics prior to Galileo emerged.

The doctrine of the scholastics, notes M Painlevé again, was by no means absurd in itself. One could even say that their precise interpretation of the principle of causation was the first that had to come to mind and be tested. Their fault was to persist and pretend to bend the facts. Keeping the rest of the scholastic ideas, the Copernicans confined themselves to taking trivial observations into account in order to modify on one point - the definition of the initial conditions - the interpretation of the principle of causality, and this modification is sufficient to give Mechanics, motionless for centuries, a prodigious impulse. Between the two schools is the intermediate school of Tycho Brahé, which rejected the notion of absolute movement: all the location of movements are equal and there is no reason to adopt one rather than the other; it is a simple matter of convention. The doctrine of Tycho Brahé was unassailable from the descriptive point of view, but not from the dynamic point of view of the search for the causes of the movements. What escaped him, and what Galileo, on the contrary, highlighted, was that the principle of causality and in particular the "corollary of symmetry" can only be checked by means of a special tracking of movements and not in any kind of tracking.

Ultimately, classical mechanics presupposes the existence of a privileged reference trihedron. This crucial point, essential for understanding the dynamics, is masterfully established by M Painlevé.

**2. Cours de Mécanique (1929), by Paul Painlevé and Charles Plâtrier.**

**2.1. Review by: Francis Dominic Murnaghan.**

*Bull. Amer. Math. Soc.*

**37**(11) (1931), 809.

This book is a printed version of lectures given by C Plâtrier (substituting for P Painlevé) to second-year students at the École Polytechnique. The main topics discussed are Rigid Dynamics, Hydrodynamics, Elasticity, Aerodynamics, Theory of Relativity. The point of view is quite modern and the treatment is very satisfactory. The discussion of the gyroscope is about the best we have seen in a general treatise as is also that on wave-propagation. The discussion of tensor analysis is adequate for a student who is not desirous of being a specialist in this subject. The only criticism that one could fairly make of the work is its very academic character; a book containing several chapters on aerodynamics which does not mention Prandtl's name cannot appeal strongly to the practical man of affairs. Nevertheless the book is a valuable addition to Appell's renowned treatise on Mechanics, which book it (together with Painlevé's Cours de Mécanique, Tome I) replaces to a certain extent.

**2.2. Review by: Haroutune Mugurditch Dadourian.**

*Bull. Amer. Math. Soc.*

**38**(7) (1932), 446-447.

This is the first volume of a series which is to present a course of lectures delivered at l'Ecole Polytechnique. The volume is divided into four books, the first of which is devoted to vector addition and multiplication. The second book is on the fundamental axioms of Newtonian mechanics, the most important of which may be stated as follows:

It is possible to adopt, once for all and for the entire universe, a measure of distance, a measure of time, and a system of reference axes such that the following principles always hold good: I. The constancy of the velocity of an isolated particle; II. The equality of action and reaction; III. The determinate character of the mutual accelerations of two isolated particles when their velocities and their distance apart at any instant are given; IV. The geometric addition of forces.

A reference system relative to which the foregoing four propositions are true is called an absolute system of axes; velocities and accelerations referred to such a system are called absolute velocities and accelerations. "The fundamental postulate of mechanics," says the author, "consists, therefore, in the admission of a system of absolute axes." Every system which has a uniform motion of translation relative to an absolute system is also an absolute system.

The third book is entitled

*The general theorems of the dynamics of systems*, and deals mainly with the motion of a particle.

The fourth book is on the general theory of the equilibrium and motion of systems. It is less elementary than the third book. D'Alembert's principle, Lagrange's equations, and the principle of virtual velocities are here applied to problems of equilibrium and of motion.

The presentation is clear, logical, and rigorous as would be expected from a French scientist of the standing of Painlevé. The book deserves a prominent place on the reference shelf of every serious student of mechanics.

The only serious criticism I would offer could be made against almost all books on mechanics. It seems to me that the presentation of Newtonian mechanics lends itself to a degree of unity and integration comparable to that of the theory of relativity. The entire subject could be based upon a single fundamental principle from which all other principles, laws, and theorems could be derived as was done by Lagrange in his Mécanique Analytique. In such a presentation the geometric addition of forces, for example, would not be elevated to the status of a fundamental principle any more than the geometric addition of other vector magnitudes, such as velocities and accelerations.

**3. Leçons sur la Résistance des Fluides non visqueux. I (1930) by Paul Painlevé.**

**3.1. Review by: Sydney Goldstein.**

*The Mathematical Gazette*

**16**(221) (1932), 360-361.

M Painlevé seeks to explain the paradox of d'Alembert, that a solid body moving with uniform velocity through a perfect fluid experiences no resistance to its motion. First, however, he will obtain general formulas for the forces and moments on a body moving in any manner whatever. Here, in this volume, he only develops the preliminaries for his theory; but he gives a summary of his results and some indications of what his theory will be. It appears that he will develop the theory of surfaces of discontinuity, and seek to explain their origin. If the velocity is anywhere discontinuous, particles originally in contact must separate. This separation takes place at the forward stagnation point, and is accompanied, according to Painlevé, by internal impulses. Indeed, M Painlevé quotes a theorem of Hadamard, that in the absence of such impulses, surfaces of discontinuity could not develop in the fluid; and presumably he will look at these impulses to explain their origin. On this preliminary announcement, two remarks are to be made. First, according to modern notions, a fluid of zero viscosity must be regarded as the limit as the limit of a fluid of small viscosity; in such a fluid, vorticity has its origin in the boundary layer, and this remains true even when the viscosity becomes zero and the boundary layer infinitely thin. The result is then the production of a vortex sheet, or surface discontinuity of velocity, in the fluid. Second, when the vorticity becomes zero, the Reynolds number becomes infinite, so that presumably the "physical" limit would be turbulent motion. Thus, however necessary it may be to stress the results that can arise from the fact that a real fluid is not a continuum in the strict mathematical sense, it is exceedingly doubtful if M Painlevé will be able to give anything like an accurate picture of physical processes.

The book, apart from its ultimate object, which is serious and forbidding, is charming. The mathematics is so clear, and the whole so well written, that it is a joy to read. The chapters on harmonic functions should give pleasure to many mathematicians, and the treatment of elementary classical hydrodynamics will appeal to all teachers of the subject. M Painleve's faith in theory is a joy to the mathematical spirit. Whether it be true or false, it is so comforting to read "On attribue communément tout l'honneur de la création des aéroplanes à l'empirisme et à l'audace des praticiens. C'est une complète erreur". And when M Painleve says "L'influence de cette viscosité qui est faible si les mouvements du liquide sont reguliers et peu turbulents au sens de M Boussinesc, devient considerable dès que l'agitation du liquide est intense, ou simplement notable; les termes du premier degré qui interviennent dans la théorie de Poiseuille sont alors considérablement dépassés par ceux du second degré qui néglige cette théorie, et dont M Boussinesc tient compte par une sorte de théorie statistique approchée" - when M Painlevé says that, the reference to Stokes's theory as the theory of Poiseuille, the notions that the influence of viscosity is small, and the theory of Stokes applicable, till the motion becomes turbulent, the light-hearted treatment of turbulence, and the omission of any other name than that of M Boussinesc, are all so delightfully naive that only a churl could find fault.

**3.2. Review by: Raymond Clarence James Howland.**

*Science Progress in the Twentieth Century (1919-1933)*

**26**(101) (1931), 151.

The first part of Prof Painlevé's lectures on the resistance of fluids has been prepared for press by M Metral. The second part, prepared by M Mazet, will follow. A large proportion of the present section consists of mathematical preparation for the main theme, and the authors' intentions have to be judged from the introduction, where a general discussion of the problem of fluid resistance is given. Professor Painlevé believes that there has been a great deal of loose thinking with regard to aeronautical theories, and he sets out to make it clear what assumptions underlie the theories of fluid resistance and how far they depend on empirical results obtained under conditions different from those to which the results are applied.

How far he has succeeded in this cannot be judged from the present volume, which is very largely concerned with such fundamental matters as the theory of harmonic functions. The kinematics of continuous media, the equations of motions of a perfect fluid, and the theory of similarity are discussed, while appendices, in addition to more analytical matter, contain a discussion of the permanence of irrotational motion; but the problem of resistance is postponed.

The style and arrangement are admirable, and the whole, though containing little that is not familiar to the advanced student of hydrodynamics, forms an excellent introduction. If the second part is worthy of this beginning, it will be of very great interest.

**4. Analyse des travaux scientifiques jusqu'en 1900 (1967), by Paul Painlevé.**

**4.1. Review by: Paul Lévy.**

*Revue de Métaphysique et de Morale*

**73**(3) (1968), 383-384.

A note from the editor tells us that this book is a reproduction of the notice written by the author, for his candidacy to the Academy of Sciences, to which he was elected in 1900, at the age of thirty-seven. This leaflet, printed in a small number of copies, is almost impossible to find today. We know, moreover, that it was his youthful work that classified Painlevé to the level of the first mathematicians of his generation (his political activity subsequently diverted him from science as well as teaching, without for that reason never becoming disinterested in it). We should add that this notice, written for scholars, most of whom were not mathematicians, is easy to read. Without doubt the formulas, essential for precise statements, are numerous. But it is not necessary to examine them very carefully to be interested in the admirable exposition of general ideas. The reading is fascinating for analysts, at least for those who do not seek to forget the mathematics from before Bourbaki.

These remarks would suffice to show the interest of the new edition undertaken by the Blanchard bookshop. Let us recall however that the most important works of Painlevé are those relating to differential equations. For a long time, analysts have known that, in general, the solutions of one of these equations are not known functions, but new functions which must be studied using this equation. After Poincaré, Painlevé made immense progress in this aspect of the theory, by studying the analytical nature of solutions and families of solutions. New and difficult problems were posed, in particular that of the distinction of the fixed singular points (that is to say the same for all the solutions) of those which are moveable, and that of the nature of the singularities (points which are critical singular or not, essential or not, the critical points can be essential or transcendent). The results obtained by Painlevé are very important; they have hardly been exceeded since then by the fine works of M R Garnier. In particular, he drew attention to new types of equations, now known as Painlevé equations I, II ..., and to new transcendent functions introduced by these equations.

Painlevé's work cannot be reduced to the study of differential equations. Its summary is only the third part of his notice. The first deals with general theory of functions, the second studies special transcendent functions, and the theory of algebraic functions of several variables, and the last with rational mechanics and celestial mechanics. In all these areas, Painlevé has obtained new and important results, which we cannot list here. In an admirable introduction (14 pages; we strongly recommend reading it by those who could not read the whole book), he explains that, despite the diversity of the subjects treated, all his works are linked to the same general idea. Knowledge of the theory of functions must precede the study of differential equations, and the results obtained on the subject of these equations apply to the equations of mechanics, the object of some of his finest works; a thesis on the n-body problem was notably honoured by the Academy (with the Bordin Prize, 1894).

**4.2. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**52**(379) (1968), 62.

Paul Painlevé (1863-1933) will doubtless continue to figure in the history books, for he was Minister for War and, for two months, Prime Minister of France, in the black days of 1917, the year of Nivelle's failure at the Chemin des Dames, of Passchendaele and Caporetto, of the French mutinies and the Russian surrender at Brest-Litovsk. As a politician he is said to have been clear-headed but too slow in making decisions.

His mathematical reputation stands high. By 1900 he had written about 100 papers, and the present booklet describes this work in detail. It was written by Painlevé himself and a few copies were privately printed in support of his claims as a candidate for the Académie des Sciences; re-discovered, it has now been made generally available for the first time. Though the work is in four sections (general function theory, special functions, differential equations, rational and celestial mechanics) differential equations form the central theme to which the other topics are closely linked.

Non-linear ordinary differential equations have been studied from Riccati onwards, cropping up in a wide variety of physical problems. Painlevé and his followers investigated the classification of non-linear equations of the second order by considering the functional character of the solution and its singularities. If the critical points, that is, branch points and essential singularities, but not poles, are fixed, some 50 types emerge, most of which can be integrated in terms of elementary functions or transcendental functions satisfying linear equations. But 6 types remain to define new transcendental functions, the Painlevé transcendents; for instance, the first of these arises from the equation $y'' = x + 6y^{2}$. A full and clear exposition for the English reader is given in Ince's classical volume, Ordinary differential equations.

One cannot claim that this volume is of the first importance. But it is full of interest in its description of the beginnings and growth of a valuable mathematical doctrine, as seen by the creator thereof.

**4.3. Review by: Editors.**

*Revue Philosophique de la France et de l'Étranger*

**157**(1967), 477.

The service of public affairs deserves the attention of mathematicians themselves. We will however be allowed to deplore here that the great Painlevé left differential equations so early for all-party cabinet whose calculus is more delicate, is less rigorous, less harmonious and, too often, less fertile.

Paul Painlevé thus left an unfinished work. Undoubtedly illustrious disciples, like Professor Garnier, so scrupulous publisher of his master's works, continued on the open path. But among the younger specialists, some such as my professor Charles Ehresman and his disciple Georges Reeb, insistently point out to beginner researchers that almost nothing is done, except a very difficult and very little studied dissertation due to the Russian academician Petrovsky and his pupil Evgenii Landis, to apply to the problems raised by Painlevé the new invariant methods of the theory of general dynamical systems.

May the reprint of a copious notice written by Paul Painlevé himself to present his work, make easier the research whose voices so authorised mark the need.

**4.4. Review by: Joseph Ehrenfried Hofmann.**

*Sudhoffs Archiv*

**52**(3) (1968), 286.

This is a reprint of the overview of the contributions of the famous mathematician Paul Painleve (1863-1933) originally published as

*Notice sur les travaux ...*(1900).

Last Updated July 2020